Well, the point is actually pretty simple. You start with an equation that works for some galaxies. Then you find a galaxy where it doesn't work - you adjust the equation and now it works again. But then you find a new galaxy where it still doesn't work, you adjust again. You try this for 5, 10, 20 galaxies, just adjust the equation - but at some point there is a limit where the equation doesn't bare any more adjustment, and you still have hundreds of other galaxies that don't work.
So, the alternative that starts being simpler is a single simple equation that works for all galaxies, but allow each galaxy to have varying amounts of stuff in it with mass, but that doesn't interact electromagnetically. Right now, this is the simplest solution we have that fits all observations well.
Is it the right answer? We won't be sure unless we can detect particles that fit the necessary characteristics, and a theory that explains the distribution of these particles in different kinds of galaxies. Unfortunately, the models we have allow these particles to be arbitrarily hard to detect, at the level that we can't really rule them out even if we had a particle accelerator the size of the Earth that didn't find them.
Now, in principle a different equation could exist that has the same solutions as the current equations where they work, and different solutions where they don't work, without adding O(number of galaxies) extra parameters. But just like the dark matter particles, unless we stumble upon it, we can't know if it exists or not.
It bears mentioning that the situation is even more constraining than this, because you're not just looking at galactic dynamics - you're looking at galaxy _cluster_ dynamics, and gravitational lensing measurements, and the CMB, and large-scale structure formation, and whatever else.
Dark matter is not Fermi's elephant, as invoked elsewhere in the thread. It's more like the story of the blind men and the elephant - except that the blind men recognise that their individual observations, taken together, admit a coherent explanation.
> Right now, this is the simplest solution we have that fits all observations well.
It doesn't though. For instance, the latest in a litany of such failures is that rotation curves are flat past a million light years [1]. There is no plausible DM distribution that could reproduce such rotation curves while being consistent with other observations.
So, the alternative that starts being simpler is a single simple equation that works for all galaxies, but allow each galaxy to have varying amounts of stuff in it with mass, but that doesn't interact electromagnetically. Right now, this is the simplest solution we have that fits all observations well.
Is it the right answer? We won't be sure unless we can detect particles that fit the necessary characteristics, and a theory that explains the distribution of these particles in different kinds of galaxies. Unfortunately, the models we have allow these particles to be arbitrarily hard to detect, at the level that we can't really rule them out even if we had a particle accelerator the size of the Earth that didn't find them.
Now, in principle a different equation could exist that has the same solutions as the current equations where they work, and different solutions where they don't work, without adding O(number of galaxies) extra parameters. But just like the dark matter particles, unless we stumble upon it, we can't know if it exists or not.